Sometimes, AI can be your friend:
An integrated amplifier increases the strength of electrical signals to reproduce sound. It combines a preamp, power amp, and other functions into a single unit.
IOW, an integrated amplifier already includes a preamp so, no, if you have an integrated amplifier you should not need a stand-alone preamp.
Back in the day, preamps amplified very small electrical signals so they could then be adequately amplified by the power amplifier. When digital players and DACs began outputting higher voltage signals, some people began using them to drive power amplifiers directly. In those cases, the only other thing needed was a method to adjust the volume, so passive volume controls were introduced and, later, on-board volume controls began to show up in DACs.
However, some listeners began to notice that their systems sounded better with a preamp in the signal reproduction chain. Why would a preamp be necessary when the DAC has enough output voltage to drive the power amp directly? It turns out that having active preamp circuitry, even in cases where the DAC output offers plenty of voltage, improves the impedance ratio between the output from the DAC to the input into the amplifier, which reduces the effects of the interconnect cables. As with most things in audio, there are trade-offs and the success of getting good sound without a preamp depends on the length of cables, the specific equipment used, as well as listener preferences. Therefore, one size does not fit all.